Local Homology and Local Cohomology

Abstract

Let $(R,\mathfrak{m})$ be a local ring, $I$ a proper ideal of $R$ and $M$ a finitely generated $R$-module of dimension $d$. We discuss the local homology modules of $H_I^d(M)$. When $M$ is Cohen-Macaulay, it is proved that $H_{\mathfrak{m}}^d(M)$ is co-Cohen-Macaulay of N.dimension $d$ and $H_d^{\underline{x}}(H_{\mathfrak{m}}^d(M))\cong \hat{M}$ where $\underline{x}=(x_1,...,x_d)$ is a system of parameters for $M$.

10 pages